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The algebraic behaviour of the two point correlation function is a tell-tale sign of a Luttinger liquid. The bond dimension introduces a length scale, after which the correlation function settles to a constant value. This bond dimension dependent correlation length is given in the inset.
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Though Kiely and Mueller calculate the superfluid fraction by considering helicity moduli, the phase diagram can also be reasonably well-described by calculating the expectation value of ladder operators. Here we can clearly observe a Mott lobe!
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Luttinger liquids are gapless critical phases - their entanglement entropy is described by and area law. In 1D this is dependent on the length of the system. For us, a finite bond dimension induces an effective length scale.
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As much as it may look like it, this distribution does not suggest Bose-Einstein behaviour - we cannot have symmetry breaking in 1D. Rather, what we are observing is a divergence in the momentum due to the critical nature of the system. The divergence, however, is tamed by the finite length scale introduced by the bond dimension.
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